Identify quadrics with contextuality fraction exceeding one fifth

Determine which hyperbolic or elliptic quadrics of the multi-qubit Pauli space have a minimum violated-context fraction strictly greater than one fifth, equivalently a noncontextuality-inequality ratio strictly below three fifths.

Background

The paper proves that every quadric of rank at least three forces every noncontextual assignment to violate at least one fifth of its contexts, so the associated classical-to-quantum ratio is at most three fifths. The four-qubit hyperbolic quadric attains this lower violation fraction exactly, but the authors leave unresolved whether other quadrics are more contextual. Heuristic upper bounds for several configurations suggest a possible violation fraction of five seventeenths, corresponding to a ratio of seven seventeenths.

References

By \cref{thm:general}, every quadric of $N\ge3$ qubits violates at least a fifth of its contexts in every noncontextual model, so its ratio is at most $3/5$; the open question is which of them have a violated fraction strictly above $1/5$, that is a ratio strictly below $3/5$. The heuristic upper bounds of for $W(7,2)$, $Q(7,2)$ and $Q(9,2)$ are all exactly $5/17$ of their lines, which leaves open a ratio as low as $7/17$.

— The Four-Qubit Hyperbolic Quadric Has Contextuality Degree 315  (2610.26465 - Li et al., 19 Oct 2026) in Section 5, Physical consequences, paragraph “A noncontextuality inequality with an exact classical bound”

Does a hyperbolic quadric $Q(2N-1,2)$ have degree $L/5$ whenever $5$ divides $L$, and are its optimal violated sets again a single orbit?

— The Four-Qubit Hyperbolic Quadric Has Contextuality Degree 315  (2610.26465 - Li et al., 19 Oct 2026) in Section 6, item 4 (“Higher rank”)